Consider two random variables $x$ and $Y$ with joint PMF $($probability Mass Function$)$ in the

table below. Figure $1$ shows $P_{xY}(x,y).$

Table $1$ Joint PMF of $x$ and $Y$ in Problem $1.$

Figure $1$ Joint PMF of $x$ and $Y($in Problem $1).$

a$)$ Find $P(x=0,Y1).$

b$)$ Find the marginal PMFs of $x$ and $Y.$

c$)$ Find $P(Y=1|x=0).$

d$)$ Are $x$ and $Y$ independent